The number of integral values of $\mathrm{\lambda }$ for which ${\mathrm{x}}^2+{\mathrm{y}}^2+\mathrm{\lambda x}+(1-\mathrm{\lambda })\mathrm{y}+5=0$ is the equation of a circle whose radius cannot exceed 5, is

$$\begin{gathered} {\mathrm{x}}^2+{\mathrm{y}}^2+\mathrm{\lambda x}+(1-\mathrm{\lambda })y+5=0, \\ \mathrm{given} \mathrm{that} {\mathrm{r}}^2\le 25\Rightarrow {\mathrm{r}}^2=\frac{{\mathrm{\lambda }}^2+{(1-\mathrm{\lambda })}^2}{4}-5\le 25 \\ 2{\mathrm{\lambda }}^2-2\mathrm{\lambda }-19\le 100 \end{gathered}$$ $\Rightarrow 2{\mathrm{\lambda }}^2-2\mathrm{\lambda }-119\le 0\Rightarrow \mathrm{\lambda }\in \left(\frac{1-\sqrt{239}}{2},\frac{1+\sqrt{239}}{2}\right)$

Integer values of $\mathrm{\lambda }$ are -7,-6,....-1,0,1,2,.......8

Number of integral values=16